Posts Tagged ‘Education’

The main reading for this study block (linked below) is a tricky and detailed account of current teaching relating to angles. It’s main findings are that not enough is done to develop the concept of ‘turn’ with children – that an angle can be defined at the amount of turn from one position to another, and that, if it is taught, the main focus is on right angled turns.

Beebots and roamers could be used in school to investigate the idea of turn, but why not use the school grounds? Create obstacle courses in the playground for children to be directed around – making sure that the turns aren’t always at right angles.

Mitchelmore and White state that children need to experience a wider range of angle concepts. They believe that teacher move too quickly on to the abstract idea of an angle – as shown here.

Their research looked at whether children could represent the movement of objects such as a door, or wheel in terms of diagrams and still understand what was happening – whether they could move from the physical to the abstract in one move.

For instance, the angle shown here could represent the movement of the blades of a pair of scissors, or the opening of a handheld fan, things that children could see happening, and represent in a diagram like this. However, children referred to these movements as ‘opening’, not ‘turning’.

Children were asked to represent the angles using bendy straws to demonstrate the movement and the associated angles. If children could see the angle of movement, and explain what was happening, the researchers were happy.

The researchers also looked at children’s ideas of slope, with the idea that this is an area that is overlooked in schools. I would consider this to be the case simply because of the difficulty of representing it as well as not always being able to see the angle that a hill slopes at, for instance.

…most students had some global concept of slope but that many did not quantify it by relating the sloping line to a fixed reference line. Unlike the wheel and door, where the second line may be suggested by the initial position and there is a global movement which can be copied, there is in fact very little to help a naïve student interpret a slope in terms of a standard angle.

We conjecture that many students have a global conception of slope as a single line and do not conceive it in terms of angles. Had the physical model of the hill consisted simply of a sloping plane without any supporting edges, it is likely that far fewer students would have indicated a standard angle interpretation.

Mitchelmore and White discuss how children find it easier to see the turns, angles and slopes when both elements are easily visible (the scissors, fan, etc.) and this is likely to be because it fits more readily to the idea of an angle as drawn above – something they are likely to encounter in class. They go on to say that, “the fact that the standard angle was used more frequently for the door and hill (where one arm must be constructed) than the wheel (where both arms must be constructed) supports the view that the crucial factor accounting for the rate of use of standard angles is the physical presence of the angle arms… 88% of the students used standard angle modelling when both lines were visible, 55% when only one was visible, and 36% when no line was visible.”

There are three main findings to this piece of research.

That angle work can be related to the everyday concepts of corner, slope and turn.

The fewer arms that are present in a particular angle context, the more that has to be constructed to bring it into relation to other angle contexts and, therefore, the more difficult it is to recognise the standard angle. It is only in exceptional cases that the relevant line has to be discovered. In most cases, it has to be invented through conscious mental activity.

That many children form a standard angle concept early, but that this concept is likely to be limited to situations where both arms of the angle are visible. If the concept is to develop into a general abstract angle concept, children will need more help than is presently given to identify angles in slope, turn and other contexts where one or both arms of the angle are not visible. The slope and turn domains are particularly important for the secondary mathematics curriculum, the former because of the frequency of angles of inclination in trigonometrical applications and the latter because it provides a valuable aid in teaching angle measurement.

Again, the more hands on practice children have at experiencing the different elements of angle, the stronger their knowledge is likely to be.

Having just finished my work with a bunch of Year 3 – who are far fussier than I remember – I feel quite pleased.

They were able to build and complete the secret construction quicker and more accurately than I first hoped, they used a good range of language – next to, on top of, it looks like a house (!) – and generally they worked well together.

I used magnetic polydron to build the shape for the secret construction again. In fact, I decided I would use the same shape entirely for the job, with the same colours too! It was clear that they had done this activity before. They knew what was expected of them – although that didn’t stop me from telling them – and how to go about getting to a satisfactory end. Certainly the secret construction is an activity that works lower down school than I am used to and I can see it being of value to them from a vocabulary, communication and shape point of view. The only problem with magnetic polydron is that currently we only have squares and equilateral triangles, which limits the number of shapes we can make. I have included some of their creativity with the equipment from the very end of the session when I let them have a little play…

During the string activity, it was clear that they had limited knowledge of shape names. They struggled to predict what shapes would be made – although a couple did correctly identify a hexagon. When a star shape was created with a heptagon inside, I already knew they wouldn’t have a clue what it was, so focussed on the outside shapes instead.

Now, because of troubles when lowering the string to the floor, we ended up with some unusual patterns around the edge. We had trapeziums and triangles… the trapezium intruiged me. So I asked them what they thought it was called. Instantly, we had the name ‘square‘, which lead me to ask why they considered it a square. The response I got showed their knowledge, but also immediately reminded them they were wrong. They said, “It has four sides… but they’re not the same size, so it can’t be a square.” Intelligent thinking! This lead another child to say, “Well it must be a rectangle then!” Prompting another to say, “But it doesn’t have four right angles.”

While I wasn’t expecting this at all, it showed that a simple thing can generate such a wonderful discussion. To me it doesn’t matter that they didn’t know what a trapezium was, it was valuable enough for me to go back to their teacher and tell him that those children knew the properties, roughly anyway, of a rectangle and a square. And thinking about it, that’s all they should know. After all, it’s the fourth week of their first half term in Key Stage 2 – their knowledge of shape hasn’t been touched since the back end of Year 2 anyway, and that would consist of looking at the names of basic shapes.

If it taught me one thing, it’s that I haven’t been around Year 3 enough lately, that I’ve become used to the language and abilities of Years 5 and 6 so much that I’ve desensitised myself from children further down the school.

I certainly need to make time to work with them more throughout this course.

I’m not entirely happy with the way yesterday went. For a start, we didn’t manage to fit in everything I had originally expected, which stops me from trying out all three activities with the small group as planned. Also, the group had grown by two to thirteen, making it a little on the large side to do much meaningful investigative work with the string.

Positives:

They all enjoyed a slightly different way of working.

They all appeared to be engaged throughout both activities.

They all gave a range of input into discussions – the outdoor environment, while far from perfect at my school – encouraged a freer feel.

It flew by. The fifty minutes scheduled for a Wednesday numeracy lesson honestly only felt like fifteen minutes.

Negatives or interesting outcomes:

My highest of high flyers really struggled with the secret construction – more on that later.

A feeling that more should have got done – did everyone make progress in that lesson? It’s hard to tell. Maybe my pacing was off.

The larger group number made it difficult to get the most from the outdoor activity.

My fabulous mathematician, the sort of child anyone would want in their lesson as a human calculator, confirmed my long help suspicions – that his mathematical talent lays mainly with number and most other aspects of the subject are weaker for him.

For instance, in the opening task, the secret construction, he failed to notice that the colours his partner was using were the backs of the magnetic pieces, therefore all black. I’ve recreated the shape they had to make and the outcome he instructed.

He used all the correct pieces, just back to front. Also, when discussing the shapes made with string, he was adamant that that a turn between two sides would be around 70° when it was an obtuse angle – something we had been discussing only he day before.

A mixed one this so far…

I am developing these ideas further today with Year 6 and have a session planned to work with some Year 3 children next week along similar lines.

I attended the first meeting of many over the next two years as I begin my professional journey to a richer, greener, hopefully much improved, teaching field. I have many ideas about what I would like to achieve from this course – specific, actual targets aren’t something I’ve thought about. Yet.

I know where I want my career to go eventually in that I’d like to be someone who is a creative teacher of maths, and other subjects. I want to be known by the people I work with as someone they can mine for ideas or suggestions. I have plenty of ambition, plenty of drive, I’m willing to try anything at least once if I think it will help my pupils get a richer educational experience – although I’m slightly skeptical before I try something without knowing it’s worked elsewhere. This reason is precisely why I’m a follower of many teaching professionals on Twitter, why I spend time reading all kinds of educational web sites and forums.

Anyway, on with the course. The idea behind the Mathematics Specialist Teacher role is to become a “Mathematics Champion”. Someone who is, according to the course handbook:

…a confident and competent mathematician who can inspire children and teachers and be truly regarded as a Champion of Mathematics in the schools in which they work.

The initial development of this role was outlined in the 2008 Williams Report (Independent Review of Mathematics Teaching in Early Years Settings and Primary Schools – WMR Final Report). Recommendation 3 of which states:

There should be at least one Mathematics Specialist in each primary school, in post within 10 years, with deep mathematical subject and pedagogical knowledge, making appropriate arrangements for small and rural schools. Implementation should commence in 2009 and be targeted initially to maximise impact on standards and to narrow attainment gaps.

Now, this is likely to go out of the window in the future as the admittedly much needed money saving cuts are put in place by the current government. Indeed, funding is in the air for future cohorts. I am part of the second such cohort in Kirklees, a member of a group of 40 individuals – but Calderdale will only have one cohort, consisting of just 10 people as their second cohort funding was not approved.